If the arbitrary function \(\tilde{f}(\hat{{v}}) \)  is a product of powers, this expression reduces to the conformal correlator \(\prod _{i<j}^n x_{ij}^{2\tilde{\alpha }_{ij}}\)  where the \(\tilde{\alpha }_{ij}\)  satisfy \(\sum _{j\ne i} \tilde{\alpha }_{ij}=-\Delta _i\) .
More generally, wherever \(\tilde{f}(\hat{{v}})\)  admits a Mellin-Barnes representation,
we recover \(\prod _{i<j}^n x_{ij}^{2\alpha _{ij}}\)  times a function of position-space cross ratios as shown in [1]}. However, the most straightforward way to check that (REF ) solves the conformal Ward identities is to note that, when acting on a function \(F=F(\lbrace  x_{kl}^2\rbrace )\)  of the squared coordinate separations,

in (2 xixixi-xi2 xi+2i xi) F =
in 2xi(i+j
